A simplified three-dimensional displacement discontinuity method for multiple fracture simulations

Abstract

The displacement discontinuity method (DDM) developed by Crouch (Int J Numer Methods Eng 10:301–343, 1976) is a popular form of the boundary element method and is widely used to model hydraulic fracture propagation. The two-dimensional displacement discontinuity method (2D DDM) has limitations with regard to the simulation of some practical fracture problems, which often require accounting for the three-dimensional (3D) nature of the fracture. A 2D method with a 3D correction factor proposed by Olson (The initiation, propagation, and arrest of joints and other fractures, vol 231. Geological Society of London Special Publication, London, pp 73–87, 2004) is able to account for 3D effects of a single fixed-height, embedded fracture. However, this correction factor proves inadequate for describing multiple fracture interaction. Greater accuracy is clearly possible with a truly three-dimensional displacement discontinuity method (3D DDM), but such an approach requires significantly higher computational time and memory, especially for simulating multiple fracture propagation. To enhance calculation efficiency and reduce memory usage, a novel, simplified 3D DDM approach is proposed. This method is simplified from 3D DDM through excluding non-vertical fractures and vertical components of shear stress, as well as eliminating the need for discretization in the vertical (height) direction by applying correction factors to improve fracture-induced stresses. The correction factors are derived from the analytical plane strain solution for a uniformly loaded, isolated, vertical fracture of finite height and infinite length. The simplified method not only can calculate displacement discontinuities and induced stresses for single 3D fracture, but also can accurately and efficiently solve the problem of multiple 3D interacting fracture, improving computational efficiency by more than one thousand times compared with the standard 3D method. This model has been used to simulate the propagation of multiple fractures for horizontal wells. The approach should be generally applicable to other 3D boundary element methods to enhance computation efficiency.

Appendix

The partial derivatives of the kernel analytical solution I for 3D displacement discontinuity methods

The partial derivatives in this section were given by Shou (1993). They are presented here for completeness.

$$\begin{aligned} I_{,1}= & {} \left. {\ln (r+x_2 -\xi _2 )} \right\| \\ I_{,2}= & {} \left. {\ln (r+x_1 -\xi _1 )} \right\| \\ I_{,3}= & {} \left. {-\tan ^{-1} \Bigg (\frac{(x_1 -\xi _1 )(x_2 -\xi _2 )}{x_3 r} \Bigg )} \right\| \\ I_{,11}= & {} \left. {\frac{(x_1 -\xi _1 )}{r(r+x_2 -\xi _2 )}} \right\| \\ I_{,22}= & {} \left. {\frac{x_2 -\xi _2 }{r(r+x_1 -\xi _1 )}} \right\| \\ I_{,33}= & {} \left. {\frac{(x_1 -\xi _1 )(x_2 -\xi _2 )(x_3^{2}+r^{2})}{r(x_3^{2}+(x_1 -\xi _1)^{2})(x_3^{2}+(x_2 -\xi _2)^{2})}} \right\| \\ I_{,12}= & {} \left. {\frac{1}{r}} \right\| \\ I_{,13}= & {} \left. {\frac{x_3 }{r(r+x_2 -\xi _2 )}} \right\| \\ I_{,23}= & {} \left. {\frac{x_3 }{r(r+x_1 -\xi _1 )}} \right\| \\ I_{,111} \!= & {} \! \left. {-\frac{(r\!+\!x_2 \!-\!\xi _2 )((x_1 \!-\!\xi _1)^{2}\!-\!r^{2})\!+\!(x_1 \!-\!\xi _1)^{2}r}{r^{3}(r\!+\!x_2 \!-\!\xi _2)^{2}}} \right\| \\ I_{,211}= & {} \left. {\frac{-(x_1 -\xi _1 )}{r^{3}}} \right\| \\ I_{,311}= & {} \left. {-\frac{(x_1 -\xi _1 )x_3 (2r+x_2 -\xi _2 )}{r^{3}(r+x_2 -\xi _2)^{2}}} \right\| \\ I_{,122}= & {} \left. {-\frac{(x_2 -\xi _2 )}{r^{3}}} \right\| \\ I_{,222} \!= & {} \! \left. {-\frac{(r\!+\!x_1 \!-\!\xi _1 )((x_2 \!-\!\xi _2)^{2}\!-\!r^{2})\!+\!(x_2 \!-\!\xi _2)^{2}r}{r^{3}(r\!+\!x_1 -\xi _1)^{2}}} \right\| \\ I_{,322}= & {} \left. {-\frac{(x_2 -\xi _2 )x_3 (2r+x_1 -\xi _1 )}{r^{3}(r+x_1 -\xi _1)^{2}}} \right\| \\ I_{,133}= & {} \left. {-\frac{(r+x_2 -\xi _2 )(x_3^{2}-r^{2})+x_3^{2}r}{r^{3}(r+x_2 -\xi _2)^{2}}} \right\| \\ I_{,233}= & {} \left. {-\frac{(r+x_1 -\xi _1 )(x_3^{2}-r^{2})+x_3^{2}r}{r^{3}(r+x_1 -\xi _1)^{2}}} \right\| \end{aligned}$$

$$\begin{aligned} I_{,333} \!= & {} \! \left. {-x_3 (x_1 \!-\!\xi _1 )(x_2 \!-\!\xi _2 )\frac{(x_3^{2}\!+\!(x_1 \!-\!\xi _1)^{2})^{2}(x_3^{2}\!+\!(x_2 \!-\!\xi _2)^{2}\!+\!2r^{2})\!+\!(x_3^{2}\!+\!(x_2 \!-\!\xi _2)^{2})^{2}(x_3^{2}\!+\!(x_1 \!-\!\xi _1)^{2}\!+\!2r^{2})}{r^{3}(x_3^{2}\!+\!(x_2 \!-\!\xi _2)^{2})^{2}(x_3^{2}\!+\!(x_1 -\xi _1)^{2})^{2}}} \right\| \\ I_{,123}= & {} \left. {-\frac{x_3 }{r^{3}}} \right\| \end{aligned}$$

The symbol $\Vert$ denotes Chinnery’s notation to represent the substitution

$$\begin{aligned}&\left. {I_{,1} (\xi _1 ,\xi _2 )} \right\| =I_{,1} (a,b)-I_{,1} (a,-b)-I_{,1} (-a,b)\\&\quad +\,\, I_{,1} (-a,-b). \end{aligned}$$

where $x_{1}, x_{2}, x_{3}$ are the local coordinate of the element center, a, b are the size of element (Fig. 7), the notation ‘,i’ (where $i = 1, 2, 3$) represents the partial derivative with respect to $x_{i}$.